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Re: 2+2=4 ... How?
Posted:
Sep 29, 2012 10:40 AM


Barb Knox wrote: > > In article <op.wk94sq2i1hq4pq@leonhp>, wilson <winslole@udayton.edu> > wrote: > > > On Thu, 27 Sep 2012 01:40:34 0400, wilson <winslole@udayton.edu> wrote: > > > > > On Thu, 27 Sep 2012 01:18:28 0400, Brian M. Scott <b.scott@csuohio.edu> > > > wrote: > > > > > >> On Wed, 26 Sep 2012 21:21:54 0700 (PDT), Madhur > > >> <madhur.varshney@gmail.com> wrote in > > >> <news:dca7ee3d94464fccb4fbc01fdbc685ea@googlegroups.com> > > >> in alt.math.undergrad: > > >> > > >>> The natural numbers that we use are said to be derived > > >>> from what so called Peano's Axioms. > > >> > > >> They *can* be; this is not the only possible formal > > >> foundation for them. > > >> > > >>> While these axioms (listed below) give a method of > > >>> building up counting numbers they do not define or > > >>> construct basic arithmetic operations like addition, > > >>> subtraction, multiplication, etc or basic comparisons > > >>> like that of equality. > > >> > > >> Equality is assumed to be a known relation. The arithmetic > > >> operations and the linear ordering on the natural numbers > > >> are defined using the axioms. This is explained, albeit > > >> briefly, in the Wikipedia article on the Peano axioms: > > >> > > >> <http://en.wikipedia.org/wiki/Peano_axioms?banner=none#Arithmetic> > > >> > > >> [...] > > >> > > >> Brian > > > > > > A bit more complicated: > > > > > > First you need another axiom. From the Wikipedia article: > > > > > > Addition is the function + : N Å~ N Å® N (written in the usual infix > > > notation, mapping elements of N to other elements of N), defined > > > recursively as: > > > > > > a + S(0) = a > > > a + S(b) = S(a+b) > > > Now can define 1 as S(0), 2 as SS()) and 4 as SSSS(0). > > > > > > the proof that 2 + 2 = 4 is then a matter of substituting the right > > > thing in the right place. > > > > > > sorry. my mistake. Define 2 as SS((0)). > > Also, a + 0 = a, instead of a + S(0) = a.
Yebbut note that Madhur's post made no reference 0, for her the natural numbers begin at 1.
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